Properties

Label 2-1568-1.1-c1-0-30
Degree $2$
Conductor $1568$
Sign $-1$
Analytic cond. $12.5205$
Root an. cond. $3.53843$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.41·5-s − 3·9-s + 1.41·13-s + 7.07·17-s − 2.99·25-s − 4·29-s − 12·37-s − 12.7·41-s + 4.24·45-s − 14·53-s − 15.5·61-s − 2.00·65-s + 15.5·73-s + 9·81-s − 10.0·85-s − 4.24·89-s − 7.07·97-s + 12.7·101-s − 20·109-s + 14·113-s − 4.24·117-s + ⋯
L(s)  = 1  − 0.632·5-s − 9-s + 0.392·13-s + 1.71·17-s − 0.599·25-s − 0.742·29-s − 1.97·37-s − 1.98·41-s + 0.632·45-s − 1.92·53-s − 1.99·61-s − 0.248·65-s + 1.82·73-s + 81-s − 1.08·85-s − 0.449·89-s − 0.717·97-s + 1.26·101-s − 1.91·109-s + 1.31·113-s − 0.392·117-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1568\)    =    \(2^{5} \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(12.5205\)
Root analytic conductor: \(3.53843\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1568} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1568,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 + 3T^{2} \)
5 \( 1 + 1.41T + 5T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 - 1.41T + 13T^{2} \)
17 \( 1 - 7.07T + 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 + 4T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + 12T + 37T^{2} \)
41 \( 1 + 12.7T + 41T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + 14T + 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 + 15.5T + 61T^{2} \)
67 \( 1 + 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 15.5T + 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + 4.24T + 89T^{2} \)
97 \( 1 + 7.07T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.940696255799733527976757968662, −8.144234687068523397466885463802, −7.64924418530679762834518186282, −6.58910494162729178610576965446, −5.67525630074849547285919795819, −4.99938843313010453410834149148, −3.63974803479058916205388777857, −3.19073958456411393365617171216, −1.64682263510376091891555248700, 0, 1.64682263510376091891555248700, 3.19073958456411393365617171216, 3.63974803479058916205388777857, 4.99938843313010453410834149148, 5.67525630074849547285919795819, 6.58910494162729178610576965446, 7.64924418530679762834518186282, 8.144234687068523397466885463802, 8.940696255799733527976757968662

Graph of the $Z$-function along the critical line